cp's OEIS Frontend

A307208 a(n) is the forgotten index of the Fibonacci cube Gamma(n).

%I A307208 #19 Mar 30 2019 08:40:00
%S A307208 2,10,52,158,466,1192,2914,6722,14972,32286,67914,139824,282754,
%T A307208 562970,1105892,2146846,4124258,7849496,14815202,27752338,51632620,
%U A307208 95465502,175508250,320981472,584214530,1058602666,1910305300,3434059166,6151218034,10981579528
%N A307208 a(n) is the forgotten index of the Fibonacci cube Gamma(n).
%C A307208 The Fibonacci cube Gamma(n) can be defined as the graph whose vertices are the binary strings of length n without two consecutive 1's and in which two vertices are adjacent when their Hamming distance is exactly 1.
%C A307208 The forgotten topological index of a simple connected graph is the sum of the cubes of its vertex degrees.
%C A307208 In the Maple program, T(n,k) gives the number of vertices of degree k in the Fibonacci cube Gamma(n) (see A245825).
%H A307208 B. Furtula and I. Gutman, <a href="https://doi.org/10.1007/s10910-015-0480-z">A forgotten topological index</a>, J. Math. Chem. 53 (4), 1184-1190, 2015.
%H A307208 S. Klavžar, <a href="http://dx.doi.org/10.1007/s10878-011-9433-z">Structure of Fibonacci cubes: a survey</a>, J. Comb. Optim., 25, 2013, 505-522.
%H A307208 S. Klavžar, M. Mollard and M. Petkovšek, <a href="https://doi.org/10.1016/j.disc.2011.03.019">The degree sequence of Fibonacci and Lucas cubes</a>, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.
%F A307208 a(n) = Sum_{k=1..n} T(n,k)*k^3 where T(n,k) = Sum_{i=0..k} binomial(n-2*i, k-i)*binomial(i+1, n-k-i+1).
%F A307208 Conjectures from _Colin Barker_, Mar 29 2019: (Start)
%F A307208 G.f.: 2*x*(1 + x + 8*x^2 - 7*x^3 + 4*x^4 - 3*x^5 + 3*x^6) / (1 - x - x^2)^4.
%F A307208 a(n) = 4*a(n-1) - 2*a(n-2) - 8*a(n-3) + 5*a(n-4) + 8*a(n-5) - 2*a(n-6) - 4*a(n-7) - a(n-8) for n>8.
%F A307208 (End)
%e A307208 a(2) = 10 because the Fibonacci cube Gamma(2) is the path-tree P_3 having 2 vertices of degree 1 and 1 vertex of degree 2; consequently, the forgotten index is 1^3 + 1^3 + 2^3 = 10.
%p A307208 T := (n,k) -> add(binomial(n-2*i, k-i)*binomial(i+1, n-k-i+1), i=0..k):
%p A307208 seq(add(T(n,k)*k^3, k=1..n), n=1..30);
%o A307208 (PARI) T(n,k) = sum(i=0, k, binomial(n-2*i, k-i)*binomial(i+1, n-k-i+1));
%o A307208 a(n) = sum(k=1, n, T(n,k)*k^3); \\ _Michel Marcus_, Mar 30 2019
%Y A307208 Cf. A245825, A306967, A307157.
%K A307208 nonn
%O A307208 1,1
%A A307208 _Emeric Deutsch_, Mar 28 2019